1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Direction of the fastest rate of change of a function (Solution inclu)

  1. Apr 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Find all the points at which the direction of the fastest rate of change of the function [itex]f(x,y)=x^{2}+y^{2}-2x-4y[/itex] is i+j



    2. Relevant equations



    3. The attempt at a solution
    The direction of the fastest change is [itex]\nabla f(x,y)=(2x-2)i+(2y-4)j[/itex], so we need to find all points (x,y) where [itex]\nabla f(x,y)[/itex] is parallel to i+j.

    [itex]\Longleftrightarrow(2x-2)i+(2y-4)j=k(i+j)[/itex] [itex]\Longleftrightarrow[/itex] [itex]k=2x-2[/itex] and [itex]k=2y-4[/itex]

    Ok here's where I start to get lost. How can k = two different things at the same time?

    The solution continues:
    Then [itex]2x-2=2y-4[/itex] [itex]\Longrightarrow y=x+1[/itex], so the direction of the fastest change is i+j at all points on the line y=x+1
     
  2. jcsd
  3. Apr 18, 2013 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Right. You can't have ##k## equal two different things. What you do have to do is figure out what relation ##y## and ##x## must satisfy so that you have only one value ##k## needed. That requires that ##2x-2=2y-4##.
     
  4. Apr 18, 2013 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    The same way 4 can be equal to 2+ 2 and 3+ 1 "at the same time"= they are different ways of writing the same thing. That is why you can, as you say, write 2x- 2= 2y- 4.

     
  5. Apr 18, 2013 #4
    Ok, it just clicked.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Direction of the fastest rate of change of a function (Solution inclu)
Loading...